Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure

One-site density matrix renormalization group and alternating minimum energy algorithm

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT) / matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, PA.C. G. Baker, U. L. Hetmaniuk, R. B. Lehoucq, and H. K. Thornquist. 2009. Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Trans. Math. Softw. 36, 3, 13:1--13:23.S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. Smith, and H. Zhang. 2011. PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.2, Argonne National Laboratory.S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. 1997. Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, Eds., Birkhaüser, 163--202.M. A. Brebner and J. Grad. 1982. Eigenvalues of Ax =λ Bx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process. Linear Algebra Appl. 43, 99--118.C. Campos, J. E. Roman, E. Romero, and A. Tomas. 2011. SLEPc users manual. Tech. Rep. DSICII/24/02 - Revision 3.2, D. Sistemes Informàtics i Computació, Universitat Politècnica de València. http://www.grycap.upv.es/slepc.T. Dannert and F. Jenko. 2005. Gyrokinetic simulation of collisionless trapped-electronmode turbulence. Phys. Plasmas 12, 7, 072309.E. R. Davidson. 1975. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 1, 87--94.T. A. Davis and Y. Hu. 2011. The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1, 1:1--1:25.H. C. Elman, A. Ramage, and D. J. Silvester. 2007. Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2. Article 14.T. Ericsson and A. Ruhe. 1980. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comp. 35, 152, 1251--1268.M. Ferronato, C. Janna, and G. Pini. 2012. Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 5, 797--815.D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst. 1998. Jacobi--Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94--125.M. A. Freitag and A. Spence. 2007. Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem. Electron. Trans. Numer. Anal. 28, 40--64.M. Genseberger. 2010. Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. App. Numer. Math. 60, 11, 1083--1099.V. Hernandez, J. E. Roman, and A. Tomas. 2007. Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33, 7--8, 521--540.V. Hernandez, J. E. Roman, and V. Vidal. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 3, 351--362.V. Heuveline, B. Philippe, and M. Sadkane. 1997. Parallel computation of spectral portrait of large matrices by Davidson type methods. Numer. Algor. 16, 1, 55--75.M. E. Hochstenbach. 2005a. Generalizations of harmonic and refined Rayleigh-Ritz. Electron. Trans. Numer. Anal. 20, 235--252.M. E. Hochstenbach. 2005b. Variations on harmonic Rayleigh--Ritz for standard and generalized eigenproblems. Preprint, Department of Mathematics, Case Western Reserve University.M. E. Hochstenbach and Y. Notay. 2006. The Jacobi--Davidson method. GAMM Mitt. 29, 2, 368--382.F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang. 2010. A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation. J. Comput. Phys. 229, 8, 2932--2947.A. V. Knyazev. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517--541.A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov. 2007. Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29, 5, 2224--2239.J. Kopal, M. Rozložník, M. Tuma, and A. Smoktunowicz. 2012. Rounding error analysis of orthogonalization with a non-standard inner product. Numer. Math. 52, 4, 1035--1058.D. Kressner. 2006. Block algorithms for reordering standard and generalized Schur forms. ACM Trans. Math. Softw. 32, 4, 521--532.R. B. Lehoucq, D. C. Sorensen, and C. Yang. 1998. ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.Z. Li, Y. Saad, and M. Sosonkina. 2003. pARMS: a parallel version of the algebraic recursive multilevel solver. Numer. Linear Algebra Appl. 10, 5--6, 485--509.J. R. McCombs and A. Stathopoulos. 2006. Iterative validation of eigensolvers: a scheme for improving the reliability of Hermitian eigenvalue solvers. SIAM J. Sci. Comput. 28, 6, 2337--2358.F. Merz, C. Kowitz, E. Romero, J. E. Roman, and F. Jenko. 2012. Multi-dimensional gyrokinetic parameter studies based on eigenvalues computations. Comput. Phys. Commun. 183, 4, 922--930.R. B. Morgan. 1990. Davidson's method and preconditioning for generalized eigenvalue problems. J. Comput. Phys. 89, 241--245.R. B. Morgan. 1991. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154--156, 289--309.R. B. Morgan and D. S. Scott. 1986. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7, 3, 817--825.R. Natarajan and D. Vanderbilt. 1989. A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices. J. Comput. Phys. 82, 1, 218--228.M. Nool and A. van der Ploeg. 2000. A parallel Jacobi--Davidson-type method for solving large generalized eigenvalue problems in magnetohydrodynamics. SIAM J. Sci. Comput. 22, 1, 95--112.J. Olsen, P. Jørgensen, and J. Simons. 1990. Passing the one-billion limit in full configuration-interaction (FCI) calculations. Chem. Phys. Lett. 169, 6, 463--472.C. C. Paige, B. N. Parlett, and H. A. van der Vorst. 1995. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 2, 115--133.E. Romero and J. E. Roman. 2011. Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi--Davidson eigensolver. Concur. Comput.: Pract. Exp. 23, 17, 2179--2191.Y. Saad. 1993. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 2, 461--469.G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. 1996. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 3, 595--633.G. L. G. Sleijpen and H. A. van der Vorst. 1996. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 2, 401--425.G. L. G. Sleijpen and H. A. van der Vorst. 2000. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42, 2, 267--293.G. L. G. Sleijpen, H. A. van der Vorst, and E. Meijerink. 1998. Efficient expansion of subspaces in the Jacobi--Davidson method for standard and generalized eigenproblems. Electron. Trans. Numer. Anal. 7, 75--89.A. Stathopoulos. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue. SIAM J. Sci. Comput. 29, 2, 481--514.A. Stathopoulos and J. R. McCombs. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: Seeking many eigenvalues. SIAM J. Sci. Comput. 29, 5, 2162--2188.A. Stathopoulos and J. R. McCombs. 2010. PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description. ACM Trans. Math. Softw. 37, 2, 21:1--21:30.A. Stathopoulos and Y. Saad. 1998. Restarting techniques for the (Jacobi-)Davidson symmetric eigenvalue methods. Electron. Trans. Numer. Anal. 7, 163--181.A. Stathopoulos, Y. Saad, and C. F. Fischer. 1995. Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 64, 3, 197--215.A. Stathopoulos, Y. Saad, and K. Wu. 1998. Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19, 1, 227--245.G. W. Stewart. 2001. Matrix Algorithms. Volume II: Eigensystems. SIAM, Philadelphia, PA.H. A. van der Vorst. 2002. Computational methods for large eigenvalue problems. In Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., Vol. VIII, Elsevier, 3--179.H. A. van der Vorst. 2004. Modern methods for the iterative computation of eigenpairs of matrices of high dimension. Z. Angew. Math. Mech. 84, 7, 444--451.T. van Noorden and J. Rommes 2007. Computing a partial generalized real Schur form using the Jacobi--Davidson method. Numer. Linear Algebra Appl. 14, 3, 197--215.T. D. Young, E. Romero, and J. E. Roman. 2013. Parallel finite element density functional computations exploiting grid refinement and subspace recycling. Comput. Phys. Commun. 184, 1, 66--72

High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality

In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context

Canonical Cortical Field Theories

We characterise the dynamics of neuronal activity, in terms of field theory, using neural units placed on a 2D-lattice modelling the cortical surface. The electrical activity of neuronal units was analysed with the aim of deriving a neural field model with a simple functional form that still able to predict or reproduce empirical findings. Each neural unit was modelled using a neural mass and the accompanying field theory was derived in the continuum limit. The field theory comprised coupled (real) Klein-Gordon fields, where predictions of the model fall within the range of experimental findings. These predictions included the frequency spectrum of electric activity measured from the cortex, which was derived using an equipartition of energy over eigenfunctions of the neural fields. Moreover, the neural field model was invariant, within a set of parameters, to the dynamical system used to model each neuronal mass. Specifically, topologically equivalent dynamical systems resulted in the same neural field model when connected in a lattice; indicating that the fields derived could be read as a canonical cortical field theory. We specifically investigated non-dispersive fields that provide a structure for the coding (or representation) of afferent information. Further elaboration of the ensuing neural field theory, including the effect of dispersive forces, could be of importance in the understanding of the cortical processing of information.Comment: 19 pages, 1 figur

Algorithms for entanglement renormalization

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.Comment: 23 pages, 28 figure

Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble

We consider the Elliptic Ginibre Ensemble, a family of random matrix models that interpolate between the Ginibre Ensemble and the Gaussian Unitary Ensemble and such that its empirical spectral measure converges to the uniform measure on an ellipse. We show the convergence in law of its normalised characteristic polynomial outside of this ellipse. Our proof contains two main steps. We first show the tightness of the normalised characteristic polynomial as a random holomorphic function using the link between the Elliptic Ginibre Ensemble and Hermite polynomials. This part relies on the uniform control of the Hermite kernel which is derived from the recent work of Akemann, Duits and Molag. In the second step, we identify the limiting object as the exponential of a Gaussian analytic function. The limit expression is derived from the convergence of traces of random matrices, based on an adaptation of techniques that were used to study fluctuations of Wigner and deterministic matrices by Male, Mingo, P{\'e}ch{\'e} and Speicher. This work answers the interpolation problem raised in the work of Bordenave, Chafa{\"i} and the second author of this paper for the integrable case of the Elliptic Ginibre Ensemble and is therefore a fist step towards the conjectured universality of this result